We show that there is a rational vector space $V$ such that, whenever $V$ is finitely coloured, there is an infinite set $X$ whose sumset $X+X$ is monochromatic. Our example is the rational vector space of dimension $\sup\{\aleph_0,2^{\aleph_0},2^{2^{\aleph_0}},\ldots\,\}$. This complements a result of Hindman, Leader and Strauss, who showed that the result does not hold for dimension below $\aleph_ω$. So our result is best possible under GCH.
N. Hindman, I. Leader and D. Strauss proved that it is consistent that there is a finite colouring of $\mathbb R$ so that no infinite sumset $X+X=\{x+y:x,y\in X\}$ is monochromatic. Our aim in this paper is to prove a consistency result in the opposite direction: we show that, under certain set-theoretic assumptions, for any $c:\mathbb R\to r$ with $r$ finite there is an infinite $X\subseteq \mathbb R$ so that $c$ is constant on $X+X$.
School of Mathematical Sciences, University of Science and Technology of China（中国科学技术大学数学科学学院）；School of Mathematical Sciences, Xiamen University（厦门大学数学科学学院）
Suppose that we have a finite colouring of the reals. What sumset-type structures can we hope to find in some colour class? One of our aims is to show that there is such a colouring for which no uncountable set has all of its pairwise sums monochromatic. We also show that there is such a colouring such that there is no infinite set X with X+X (the pairwise sums from X, allowing repetition) monochromatic. These results assume CH. In the other direction, we show that if each colour class is measurable, or each colour class is Baire, then there is an infinite set X (and even an uncountable X, of …
We prove that in every $2$-colouring of the edges of $K_\mathbb{N}$ there exists a monochromatic infinite path $P$ such that $V(P)$ has upper density at least ${(12+\sqrt{8})}/{17} \approx 0.87226$ and further show that this is best possible. This settles a problem of Erdős and Galvin.
Extending a result of Rado to hypergraphs, we prove that for all $s, k, t \in \mathbb{N}$ with $k \geq t \geq 2$, the vertices of every $r = s(k-t+1)$-edge-coloured countably infinite complete $k$-graph can be partitioned into the cores of at most $s$ monochromatic $t$-tight Berge-paths of different colours. We further describe a construction showing that this result is best possible.
We show the existence of several infinite monochromatic patterns in the integers obtained as values of suitable symmetric polynomials. The simplest example is the following. For every finite coloring of the natural numbers $\mathbb{N}=C_1\cup\ldots\cup C_r$, there exists an increasing sequence $a<b<c<\ldots$ such that all elements below are monochromatic, that is, they belong to the same $C_i$: $$a,b,c,\ldots, a+b+ab, a+c+ac, b+c+bc,\ldots,a+b+c+ab+ac+bc+abc,\ldots.$$ The proofs use algebra in the space of ultrafilters $β\mathbb{Z}$.
For any subset $A \subseteq \mathbb{N}$, we define its upper density to be $\limsup_{ n \rightarrow \infty } |A \cap \{ 1, \dotsc, n \}| / n$. We prove that every $2$-edge-colouring of the complete graph on $\mathbb{N}$ contains a monochromatic infinite path, whose vertex set has upper density at least $(9 + \sqrt{17})/16 \approx 0.82019$. This improves on results of Erdős and Galvin, and of DeBiasio and McKenney.
In 1978, Richard Rado showed that every edge-coloured complete graph of countably infinite order can be partitioned into monochromatic paths of different colours. He asked whether this remains true for uncountable complete graphs and a notion of \emph{generalised paths}. In 2016, Daniel Soukup answered this in the affirmative and conjectured that a similar result should hold for complete bipartite graphs with bipartition classes of the same infinite cardinality, namely that every such graph edge-coloured with $r$ colours can be partitioned into $2r-1$ monochromatic generalised paths with each …
Let $\vec{K}_{\mathbb{N}}$ be the complete symmetric digraph on the positive integers. Answering a question of DeBiasio and McKenney, we construct a 2-colouring of the edges of $\vec{K}_{\mathbb{N}}$ in which every monochromatic path has density 0. On the other hand, we show that, in every colouring that does not have a directed path with $r$ edges in the first colour, there is directed path in the second colour with density at least $\frac1r$.
Given a finite coloring (or finite partition) of the free semigroup $A^+$ over a set $A$, we consider various types of monochromatic factorizations of right sided infinite words $x\in A^ω$. Some stronger versions of the usual notion of monochromatic factorization are introduced. A factorization is called sequentially monochromatic when concatenations of consecutive blocks are monochromatic. A sequentially monochromatic factorization is called ultra monochromatic if any concatenation of arbitrary permuted blocks of the factorization has the same color of the single blocks. We establish links, a…
An $r$-edge coloring of a graph or hypergraph $G=(V,E)$ is a map $c:E\to \{0, \dots, r-1\}$. Extending results of Rado and answering questions of Rado, Gyárfás and Sárközy we prove that (1.) the vertex set of every $r$-edge colored countably infinite complete $k$-uniform hypergraph can be partitioned into $r$ monochromatic tight paths with distinct colors (a tight path in a $k$-uniform hypergraph is a sequence of distinct vertices such that every set of $k$ consecutive vertices forms an edge), (2.) for all natural numbers $r$ and $k$ there is a natural number $M$ such that the vertex set of ev…
Suppose that P is an infinite set of primes such that P = A + B + C, where A,B,C are sets with at least two elements. We show that if P(x) > c x/log^d x (where P(x) = the number of elements of P that are <= x), and if A,B,C is a "regular" triple of sets, then either |A+B| <= d, or |B+C| <= d, or |A+C| <= d.
The upper density of an infinite graph $G$ with $V(G) \subseteq \mathbb{N}$ is defined as $\overline{d}(G) = \limsup_{n \rightarrow \infty}{|V(G) \cap \{1,\ldots,n\}|}/{n}$. Let $K_{\mathbb{N}}$ be the infinite complete graph with vertex set $\mathbb{N}$. Corsten, DeBiasio, Lamaison and Lang showed that in every $2$-edge-colouring of $K_{\mathbb{N}}$, there exists a monochromatic path with upper density at least $(12 + \sqrt{8})/17$, which is best possible. In this paper, we extend this result to $k$-edge-colouring of $K_{\mathbb{N}}$ for $k \ge 3$. We conjecture that every $k$-edge-coloured $…
Our aim in this paper is to show that, for any $k$, there is a finite colouring of the set of rationals whose denominators contain only the first $k$ primes such that no infinite set has all of its finite sums and products monochromatic. We actually prove a `uniform' form of this: there is a finite colouring of the rationals with the property that no infinite set whose denominators contain only finitely many primes has all of its finite sums and products monochromatic. We also give various other results, including a new short proof of the old result that there is a finite colouring of the natu…
For any countably infinite graph $G$, Ramsey's theorem guarantees an infinite monochromatic copy of $G$ in any $r$-coloring of the edges of the countably infinite complete graph $K_\mathbb{N}$. Taking this a step further, it is natural to wonder how "large" of a monochromatic copy of $G$ we can find with respect to some measure -- for instance, the density (or upper density) of the vertex set of $G$ in the positive integers. Unlike finite Ramsey theory, where this question has been studied extensively, the analogous problem for infinite graphs has been mostly overlooked. In one of the few resu…
Given a coloring of the k-element subsets of an uncountable separable metric space, we show that there exists an infinite monochromatic subset which contains its limit point.
P. Erdős proved that every 2-edge coloured complete graph on the natural numbers can be vertex decomposed into two monochromatic paths of different colour. This result was extended by R. Rado to an arbitrary finite number of colours. We prove that the vertices of every finite-edge coloured infinite complete graph can be partitioned into disjoint monochromatic paths of different colours. This answers a question of R. Rado from 1978.
A matrix A is image partition regular over Q provided that whenever Q - {0} is finitely coloured, there is a vector x with entries in Q - {0} such that the entries of Ax are monochromatic. It is kernel partition regular over Q provided that whenever Q - {0} is finitely coloured, the matrix has a monochromatic member of its kernel. We establish a duality for these notions valid for both finite and infinite matrices. We also investigate the extent to which this duality holds for matrices partition regular over proper subsemigroups of Q.
Erdős conjectured that for any set $A\subseteq \mathbb{N}$ with positive lower asymptotic density, there are infinite sets $B,C\subseteq \mathbb{N}$ such that $B+C\subseteq A$. We verify Erdős' conjecture in the case that $A$ has Banach density exceeding $\frac{1}{2}$. As a consequence, we prove that, for $A\subseteq \mathbb{N}$ with positive Banach density (a much weaker assumption than positive lower density), we can find infinite $B,C\subseteq \mathbb{N}$ such that $B+C$ is contained in the union of $A$ and a translate of $A$. Both of the aforementioned results are generalized to arbitrary …
In 2006 T. Brown asked the following question: Given a non-periodic infinite word $x=x_1x_2x_3\cdots$ with values in a non-empty set $\mathbb{A},$ does there exist a finite coloring $φ: \mathbb{A}^+\rightarrow C$ relative to which $x$ does not admit a $φ$-monochromatic factorisation, i.e., a factorisation of the form $x=u_1u_2u_3\cdots$ with $φ(u_i)=φ(u_j)$ for all $i,j\geq 1$? Various partial results in support of an affirmative answer to this question have appeared in the literature in recent years. In particular it is known that the question admits an affirmative answer for all non-uniforml…
An open conjecture of Erdős states that for every positive integer $k$ there is a (least) positive integer $f(k)$ so that whenever a tournament has its edges colored with $k$ colors, there exists a set $S$ of at most $f(k)$ vertices so that every vertex has a monochromatic path to some point in $S$. We consider a related question and show that for every (finite or infinite) cardinal $κ>0$ there is a cardinal $ λ_κ$ such that in every $κ$-edge-coloured tournament there exist disjoint vertex sets $K,S$ with total size at most $ λ_κ$ so that every vertex $ v $ has a monochromatic path of length a…
We prove that for any $r\in \mathbb{N}$, there exists a constant $C_r$ such that the following is true. Let $\mathcal{F}=\{F_1,F_2,\dots\}$ be an infinite sequence of bipartite graphs such that $|V(F_i)|=i$ and $Δ(F_i)\leq Δ$ hold for all $i$. Then in any $r$-edge coloured complete graph $K_n$, there is a collection of at most $\exp(C_rΔ)$ monochromatic subgraphs, each of which is isomorphic to an element of $\mathcal{F}$, whose vertex sets partition $V(K_n)$. This proves a conjecture of Corsten and Mendonça in a strong form and generalizes results on the multicolour Ramsey numbers of bounded-…
A particular case of the Hindman--Galvin--Glazer theorem states that, for every partition of an infinite abelian group $G$ into two cells, there will be an infinite $X\subseteq G$ such that the set of its finite sums $\{x_1+\cdots+x_n\big|n\in\mathbb N\wedge x_1,\ldots,x_n\in X\text{ are distinct}\}$ is monochromatic. It is known that the same statement is false, in a very strong sense, if one attempts to obtain an uncountable (rather than just infinite) $X$. On the other hand, a recent result of Komjáth states that, for partitions into uncountably many cells, it is possible to obtain monochro…
A factorisation $x = u_1 u_2 \cdots$ of an infinite word $x$ on alphabet $X$ is called `monochromatic', for a given colouring of the finite words $X^*$ on alphabet $X$, if each $u_i$ is the same colour. Wojcik and Zamboni proved that the word $x$ is periodic if and only if for every finite colouring of $X^*$ there is a monochromatic factorisation of $x$. On the other hand, it follows from Ramsey's theorem that, for \textit{any} word $x$, for every finite colouring of $X^*$ there is a suffix of $x$ having a monochromatic factorisation. A factorisation $x = u_1 u_2 \cdots$ is called `super-monoc…
A finite or infinite matrix A with rational entries is called partition regular if whenever the natural numbers are finitely coloured there is a monochromatic vector x with Ax=0. Many of the classical theorems of Ramsey Theory may naturally be interpreted as assertions that particular matrices are partition regular. In the finite case, Rado proved that a matrix is partition regular if and only it satisfies a computable condition known as the columns property. The first requirement of the columns property is that some set of columns sums to zero. In the infinite case, much less is known. There …
